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This question comes from my notes, heavily edited, thus slightly unusual structure.

For Lie groups one can reformulate character theory as saying that

C ⊗ K(G\ pt) = C[T/W] = C[ X* ]W

where G is the complex Lie group, W its Weyl group, T its torus. (subquestion: is this correct?)

For example, one can write the character theory of a torus and SL2 as

K(Gm\ pt) = Z[q, q-1]     and     K(SL2 \ pt) = Z[q + q-1].

Question: it's interesting that we're allowed to write Z in the examples above; I wonder if this works for any G in the formula above, e.g. if the isomorphism is valid over Z or Q rather then C?

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The isomorphism works over Z. The proof is that the basis change between W-symmetrized monomials and characters is upper-triangular with 1's on the diagonal.

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  • $\begingroup$ I think I finally understand what canonical basis is, it's the generalization of this thing. Thanks! $\endgroup$
    – Ilya Nikokoshev
    Commented Nov 1, 2009 at 23:46
  • $\begingroup$ Strangely, I don't see Scott Carnahan's answer, even though the answer with the text is prominently displayed in my "recent" tab. $\endgroup$
    – Ilya Nikokoshev
    Commented Nov 1, 2009 at 23:54
  • $\begingroup$ He deleted it. My guess is he misread the question; it's very unclear to me what connection you (or he) see between this question and the theory of canonical bases. $\endgroup$
    – Ben Webster ♦
    Commented Nov 2, 2009 at 0:12
  • $\begingroup$ Ok! Then it's probably a SE bug that the name stays in the recent list and in the question. $\endgroup$
    – Ilya Nikokoshev
    Commented Nov 2, 2009 at 1:48

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